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## Evidential decision theory

This is part of the sequence, “An introduction to decision theory”. It is not designed to make sense as  a stand alone post.

You have rebuilt the artificial creature and now you’ve placed it back in the false environment. For a while, it sits and simply gathers data about the world in the form of probabilistic relationships. It observes ten creatures going to a substantial patch of food, eight of them get picked off by predators. It observes ten creatures going to a less substantial but more sheltered patch of food. None of them get eaten. It begins to form opinions about probabilities relating the predators and the two patches.

Eventually though, it grows hungry and has to choose which patch to go to. The previous incarnation of the creature just divided the world into the possibilities it survived and the possibilities it didn’t, without taking into account the effect of its decisions on its chances of survival. This new version of the creature will not fall into the same trap.

There are two ways it could have been programmed to avoid this. It could have been given a causal decision theory so it would ask  what its actions were likely to cause (would going to this patch of food cause me to be more likely to die). Causation is difficult though. The creature watches and takes in probabilistic information but this isn’t enough – there are many occasions where purely probabilistic information isn’t enough to identify a single causal structure for the world. Add in temporal information and more cases can be distinguished. However, an unobserved third variable might be responsible for the probabilistic relationships and even temporality won’t always help to distinguish this.

Causation is difficult and if instead you could read off useful information directly from just the probabilistic information, surely that would be preferred. Correlation, it turns out, is much easier to figure out. And correlation is what fuels evidential decision theory (EDT).  Naive evidential decision theory (we’ll discuss more sophisticated versions in later posts) simply looks at the correlations between the decision and the state of the world. It seems that its simple nature may give it an initial benefit as a decision theory – if evidential decision theory can do all that causal decision theory can, then, the argument goes, it wins because it has less conceptual baggage.

So how does EDT capture correlation. Let’s look at the original equation for expected utility:

$Expected \ Utility (Decision) =\sum_{i}Probability(WorldState_{i})\times Utility(WorldState_{i}\ \wedge \ Decision )$

In the previous post, we realised that the probability of the world state can’t be treated as being independent of the decision. In other words, the following term of the equation needs to be changed.

$Probability(WorldState_{i})$

Evidential decision theory does this by replacing this with the probability of the world state given the decision – $P(A \ \mid \ B)$ means the probability of A given B.

$Probability(WorldState_{i} \ \mid \ Decision)$

So evidential decision theory calls for an agent to make the decision which maximises the following formula:

$Expected \ Utility (Decision) =\sum_{i}(WorldState_{i} \ \mid \ Decision)\times Utility(WorldState_{i}\ \wedge \ Decision )$

How does this work? Well think of the patches of food. Given that the relationship used in this equation is simple correlation, the agent can easily work out that the probability of being eaten given the decision to go to the more substantial patch is much higher than the probability of being eaten given the choice to go to the less substantial patch. So it will choose to go to the less substantial patch.

To put the numbers in, we need some probabilities. Presuming the agent drew these from its earlier observations, then the probability of death given the substantial patch is 0.8 (8 out of ten creatures were eaten). On the other hand, its probability of being eaten given going to the less substantial patch is 0.

However, while we’ve been focusing on the probabilities in the last few posts (as these are the issue of the debate we’re exploring), the equation above does also mention another factor – the utility received from the combination of a world state and a decision. In a previous post we represented that in this utility table:

 Death Survival Substantial patch -10 5 Less substantial patch -10 2

Now we have all the information we need to do the necessary calculations (click on the image for a larger copy)

So evidential decision theory reaches the correct decision in the case facing our agent and it does so without relying on any complex causal apparatus. The next few posts will explore how causal decision theory reaches the same decision in at least this instance. The question that will then be asked is, does causal decision theory have advantages such that taking on the extra causal baggage is worthwhile?